Question 1204192
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Answer:  <font color=red size=4>0.8944</font> (choice A)



Explanation


We have a binomial probability distribution because of these reasons:<ul><li>There are two options: the item is defective or it isn't.</li><li>Each  item has the same chance of being defective (5% chance)</li><li>Each item is independent of one another.</li></ul>Binomial probability distribution formula
B(x) = (nCx)*(p^x)*(1-p)^(n-x)


The nCx refers to the nCr combination formula.
n = 75 items total = sample size
p = 0.05 = probability of a defective item
x = values chosen from the set {0,1,2,3,...,74,75}


Use the binomial formula to compute B(0), which is the probability of getting 0 defective items in a batch of 75.
B(x) = (nCx)*(p^x)*(1-p)^(n-x)
B(0) = (75C0)*(0.05^0)*(1-0.05)^(75-0)
B(0) = (1)*(0.05^0)*(1-0.05)^(75-0)
B(0) = 0.021344 approximately


Repeat for x = 1
B(x) = (nCx)*(p^x)*(1-p)^(n-x)
B(1) = (75C1)*(0.05^1)*(1-0.05)^(75-1)
B(1) = (75)*(0.05^1)*(1-0.05)^(75-1)
B(1) = 0.084252 approximately


B(0)+B(1) = 0.021344+0.084252 = 0.105596 represents the probability of getting 0 defective items or 1 defective item.


The probability of getting 2 or more defective items is 1 - 0.105596 = 0.894404 which rounds to <font color=red size=4>0.8944</font>
I'm using the idea that P(0 or 1 defects) + P(2 or more defects) = 1.


There's about an 89.44% chance of getting at least two defective items.


Another approach would be to use a calculator like this
<a href = "https://www.gigacalculator.com/calculators/binomial-probability-calculator.php">https://www.gigacalculator.com/calculators/binomial-probability-calculator.php</a>


Or you could use a TI83/TI84 calculator
<a href = "https://www.statology.org/binomial-probabilities-ti-84-calculator/">https://www.statology.org/binomial-probabilities-ti-84-calculator/</a>
The command you'll want is called BinomCDF.
More specifically you'll input <font color=red>1-BinomCDF(75,0.05,1)</font>


Or you could use the BinomDist spreadsheet function.


The spreadsheet command is
<font color=red>=Round(1-BinomDist(1,75,0.05,1),4)</font>
The template for BinomDist is:
BinomDist(X,N,P,C)
where
X = number of successes
N = sample size
P = probability of success
C = either 0 or 1 depending if you want a PDF or CDF
Do not forget about the equal sign up front when typing in  <font color=red>=Round(1-BinomDist(1,75,0.05,1),4)</font> or the command won't execute.


Computing BinomDist(1,75,0.05,1) will produce the value B(0)+B(1) = 0.105596 that we found earlier.
Then we subtract it from 1 to get the value of B(2)+B(3)+...+B(74)+B(75) = <font color=red size=4>0.8944</font>
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